Conjecture 1.19 — (The volume conjecture, [198, 296]) For any knot K, 2π·lim N →∞ log|JN (K)| N = v3||S3− K||, (2) where||·||denotes th…

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

A knot or link is a smooth oriented embedding of one or more circles in the oriented $3$-sphere, considered up to orientation-preserving ambient isotopy. Write $O$ for the unknot, which bounds an embedded disk. Write $J_N(K)$ for the Kashaev invariant $\langle K\rangle_N$, equivalently the reduced $N$-dimensional colored Jones polynomial evaluated at $q=\exp(2\pi i/N)$, with $J_N(O)=1$. Reduction removes the unknot quantum-dimension factor before evaluation at the root of unity. For the compact knot exterior $X_K=S^3\setminus\operatorname{int}\nu K$, the simplicial volume $\|X_K,\partial X_K\|$ is the infimum of sums of absolute coefficients of real relative singular cycles representing its oriented relative fundamental class. Let $v_3$ be the volume of a regular ideal hyperbolic tetrahedron.

Statement recorded in the supplied source.

For every knot $K$, does the limit exist and satisfy \[2\pi\lim_{N\to\infty}\frac{\log|J_N(K)|}{N} =v_3\|X_K,\partial X_K\|?\] For a hyperbolic knot the right side is its complement's complete hyperbolic volume. The recorded conjecture applies to all knots, not only hyperbolic knots. [S2]

Short English statement

Does the exponential growth of the root-of-unity colored Jones invariant recover the knot exterior simplicial volume for every knot?

Sources

• S1 ULAM. AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10400019 (AMR-103-0019). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 T. Ohtsuki (editor), Problems on invariants of knots and 3-manifolds, Geometry & Topology Monographs 4 (2002), pp. 377--572 (published 2004), item 1.19 and its surrounding definitions and remarks. https://arxiv.org/abs/math/0406190v1.